HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  brafnmul Structured version   Unicode version

Theorem brafnmul 23454
Description: Anti-linearity property of bra functional for multiplication. (Contributed by NM, 31-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
brafnmul  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( bra `  ( A  .h  B )
)  =  ( ( * `  A ) 
.fn  ( bra `  B
) ) )

Proof of Theorem brafnmul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hvmulcl 22516 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
2 brafval 23446 . . 3  |-  ( ( A  .h  B )  e.  ~H  ->  ( bra `  ( A  .h  B ) )  =  ( x  e.  ~H  |->  ( x  .ih  ( A  .h  B ) ) ) )
31, 2syl 16 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( bra `  ( A  .h  B )
)  =  ( x  e.  ~H  |->  ( x 
.ih  ( A  .h  B ) ) ) )
4 cjcl 11910 . . . 4  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
5 brafn 23450 . . . 4  |-  ( B  e.  ~H  ->  ( bra `  B ) : ~H --> CC )
6 hfmmval 23242 . . . 4  |-  ( ( ( * `  A
)  e.  CC  /\  ( bra `  B ) : ~H --> CC )  ->  ( ( * `
 A )  .fn  ( bra `  B ) )  =  ( x  e.  ~H  |->  ( ( * `  A )  x.  ( ( bra `  B ) `  x
) ) ) )
74, 5, 6syl2an 464 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( * `  A )  .fn  ( bra `  B ) )  =  ( x  e. 
~H  |->  ( ( * `
 A )  x.  ( ( bra `  B
) `  x )
) ) )
8 his5 22588 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  ~H  /\  B  e.  ~H )  ->  (
x  .ih  ( A  .h  B ) )  =  ( ( * `  A )  x.  (
x  .ih  B )
) )
983expa 1153 . . . . . 6  |-  ( ( ( A  e.  CC  /\  x  e.  ~H )  /\  B  e.  ~H )  ->  ( x  .ih  ( A  .h  B
) )  =  ( ( * `  A
)  x.  ( x 
.ih  B ) ) )
109an32s 780 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  ( A  .h  B
) )  =  ( ( * `  A
)  x.  ( x 
.ih  B ) ) )
11 braval 23447 . . . . . . 7  |-  ( ( B  e.  ~H  /\  x  e.  ~H )  ->  ( ( bra `  B
) `  x )  =  ( x  .ih  B ) )
1211adantll 695 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  B ) `  x
)  =  ( x 
.ih  B ) )
1312oveq2d 6097 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( ( * `
 A )  x.  ( ( bra `  B
) `  x )
)  =  ( ( * `  A )  x.  ( x  .ih  B ) ) )
1410, 13eqtr4d 2471 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  ( A  .h  B
) )  =  ( ( * `  A
)  x.  ( ( bra `  B ) `
 x ) ) )
1514mpteq2dva 4295 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( x  e.  ~H  |->  ( x  .ih  ( A  .h  B ) ) )  =  ( x  e.  ~H  |->  ( ( * `  A )  x.  ( ( bra `  B ) `  x
) ) ) )
167, 15eqtr4d 2471 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( ( * `  A )  .fn  ( bra `  B ) )  =  ( x  e. 
~H  |->  ( x  .ih  ( A  .h  B
) ) ) )
173, 16eqtr4d 2471 1  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( bra `  ( A  .h  B )
)  =  ( ( * `  A ) 
.fn  ( bra `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988    x. cmul 8995   *ccj 11901   ~Hchil 22422    .h csm 22424    .ih csp 22425    .fn chft 22445   bracbr 22459
This theorem is referenced by:  cnvbramul  23618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-hilex 22502  ax-hfvmul 22508  ax-hfi 22581  ax-his1 22584  ax-his3 22586
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058  df-cj 11904  df-re 11905  df-im 11906  df-hfmul 23237  df-bra 23353
  Copyright terms: Public domain W3C validator